(54 ANALYTICAL MECHANICS
In order to express equation (4) in terms of rectangular coordinates we replace tan 0 by (f[ and oh turn
But ds2=dx2+dy2, therefore eliminating dx between this equation and equation (5) and separating the variables
and then integrating
where a = — and c is the constant of integration.
Let the o>axis be so chosen that when .s -. o, // «, then c = 0. Therefore
j/ = V$H-a"2", or *s* ~ \ //- — n\ (7)
Differentiating equation (7), squaring and replacing (/«- by we have
Solving for <fa,
(8)
_ __..... a%
where i - V-1. Integrating equation (8) we get
ix t/
— « cos"1'7